Best Answer CaptainEd, 30 January 2013 - 05:06 PM

Spoiler for I would like to draw a good picture

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I'm not happy with my pictures, they do not capture my tictactoe feeling

The only 3-chip combinations that add to 15:

The A and B below indicate that, at this stage of the game, both A and B have this possibility available to them. As a player chooses one of the chips, the other player loses that possibility.

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A A A A

B B B B B B B B

Note that there are four combinations with 5, three combinations with each of 2,4,6,8, and two combinations with each of 1,3,7,9

(1) 5, A chooses 5, because it has the most combinations

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A A A A

B B B B

(2) B chooses 6, (it knocks out 3 of A's combinations--I believe 4 would have been equivalent)

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A

B B B B

(3) A chooses 4, because it knocks out half of B's possibilities, while advancing two of A's own possibilities

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A

B B

At this point, A has 54 and B has 6.

B had better not choose 7, appealing though it be, as A is forced to choose 2, which creates a fork, with A threatening both 258 and 249. I believe the choice of 2 has the similar failure.

B's better choice is 8.

(4) B chooses 8, threatening 168

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A

B B

(5) A has no immediate win, so is forced to choose 1, threatening 159

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A

B

A has 145, B has 68

(6) B is forced to choose 9

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A

B

A has 145, B has 689

(7) A chooses 7, threatening 357 and blocking B's last combination (267)

(8) B chooses 3, blocking A's last combination (357)

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After the first three moves (A5, B6, A4), I think it is clear that both players can force a draw (or can mess up and lose)

One could question whether one of the three first moves could be replaced by more powerful ones. But each move seems to have a good rationale...

The only 3-chip combinations that add to 15:

The A and B below indicate that, at this stage of the game, both A and B have this possibility available to them. As a player chooses one of the chips, the other player loses that possibility.

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A A A A

B B B B B B B B

Note that there are four combinations with 5, three combinations with each of 2,4,6,8, and two combinations with each of 1,3,7,9

(1) 5, A chooses 5, because it has the most combinations

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A A A A

B B B B

(2) B chooses 6, (it knocks out 3 of A's combinations--I believe 4 would have been equivalent)

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A

B B B B

(3) A chooses 4, because it knocks out half of B's possibilities, while advancing two of A's own possibilities

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A A A

B B

At this point, A has 54 and B has 6.

B had better not choose 7, appealing though it be, as A is forced to choose 2, which creates a fork, with A threatening both 258 and 249. I believe the choice of 2 has the similar failure.

B's better choice is 8.

(4) B chooses 8, threatening 168

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A

B B

(5) A has no immediate win, so is forced to choose 1, threatening 159

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A A A

B

A has 145, B has 68

(6) B is forced to choose 9

1 1 2 2 2 3 3 4

6 5 6 5 4 5 4 5

8 9 7 8 9 7 8 6

A

B

A has 145, B has 689

(7) A chooses 7, threatening 357 and blocking B's last combination (267)

(8) B chooses 3, blocking A's last combination (357)

-----

After the first three moves (A5, B6, A4), I think it is clear that both players can force a draw (or can mess up and lose)

One could question whether one of the three first moves could be replaced by more powerful ones. But each move seems to have a good rationale...